Homework 2 Solutions
1. Frontal motion
a. Describe the typical motion of a cold front and a warm front. How can knowledge of the
pressure tendency on a surface map help determine the motion of fronts?
Answer: Cold fronts most often extend from the SW quadrant of a low pressure system, but may extend
from the W or NW quadrant. Active cold fronts move toward the SE at an average of speed of 15-25 kts.
Depending on the length of the cold front, portions of the front may move toward the E, while other
portions move S. The portion nearest the low pressure system may be elongated to move NE with the
movement of the low pressure system. Warm fronts most often extend from the NE through SE quadrant
of a low-pressure system. They generally move toward the NE at an average speed of 10 kts. Since rising
(falling) pressure is associated with the cold (warm) side of a front, the pressure tendency can provide an
indication of the motion of a given frontal surface.
b. What are the major factors that influence frontal motion?
Answer: Generally, fronts tend to follow the motion of the upper level geostrophic wind and the direction
of the low pressure system as indicated above. This implies that frontal motion is governed by the
Coriolis force and the pressure gradient force. Also, as mentioned above, frontal motion is strongly
guided by the movement of the low-pressure system. This implies that fronts tend to move in the direction
of the greatest pressure falls.
c. Explain why cold fronts tend to move faster than warm fronts. How can this process lead
to occlusions?
Answer: Warm fronts tend to move slower than cold fronts because cold fronts tend to be stronger than
warm fronts. Moreover, warm fronts speed up (slow down) during the day (night) due to heating
(cooling). In the day, turbulent mixing due to solar radiation tends to occur on both sides of the front. At
night, however, radiational cooling creates cool, dense surface air behind the warm front, which tends to
inhibit its forward progress.
2. Frontogenesis and cyclogenesis
a. Explain how frontogenesis can occur in the developing and mature stages of a lowpressure system.
Answer: In the developing and/or mature stages of a low-pressure system, frontogenesis can occur if the
following ingredients are present: (i) westward tilted low pressure system; (ii) strong low-level speed
convergence within the warm sector of the storm; (iii) strong vorticity within the midtroposphere to west
of the surface low.
b. Explain how frontogenesis can occur in regions with weak low-pressure systems.
Answer: In regions of a low-pressure system, frontogenesis can occur if the following ingredients are
present: (i) confluent flow around 700 mb in advanced of a positively tilted trough (ii) deformation and
convergence creating frontogenetic forcing north of the front; (iii) sufficient moisture and low-level
baroclinicity.
3. Conceptual models of synoptic fronts
a. Discuss the similarities and differences between the air mass model of fronts and the
conveyor belt model of fronts.
Answer: In the air mass model of fronts, the atmosphere is a juxtaposition of air masses, separated by
boundaries where the temperature changes greatly over a short horizontal distance called fronts.
Generally, a leading warm front is followed by a faster cold front and as the cold front outpaces the
forward motion of the warm front, they collide and form an occluded front. In the air mass model, each
front is viewed as a sloping surface and motions near fronts are forced upward or downward along the
sloping frontal surfaces.
Unlike the air mass model, the conveyor belt view provides for narrow streams of air flowing along
surfaces of constant potential temperature where the airstreams represent air trajectories through the
moving frontal system. Consequently, the airstream, as they move along the frontal surface, produce
clouds and precipitation structures during various stages of midlatitude cyclone evolution. The conveyor
belt model consists of a warm conveyor belt, a cold conveyor belt, and a dry, descending airstream. The
warm conveyor belt usually transports warm air poleward and is the warm, moist air mass that is lifted
over the warm front. The cold conveyor runs parallel to the warm front on its cold side. The dry airstream
supplies drier, descending air behind the surface cold front, which may then ascend around the cyclone
center. This description implies that the conveyor belt model gives a three-dimensional view of frontal
structure.
b. In terms of the conveyor belt model for fronts, explain the structure of the katafront and
anafront. How are they similar and how do they differ? Your discussion should include
the role of the warm conveyor belt, the role of the dry intrusion, and the resulting
precipitation from each type of cold front.
Answer: For an ana cold front, warm air ascends up the frontal surface to very high altitudes due to strong
convergence within the baroclinic zone between the two air masses on the frontal boundary. This causes
the frontal cloud band and the warm conveyor belt to appear behind the surface front with a dry intrusion
parallel to the warm conveyor belt, leading to strong precipitation behind the front with a sharp rear cloud
edge of frontal cloudiness. Conversely, for a kata cold front, the warm air ascent is restricted by the dry
intrusion originating behind the front and increasing strong stability. This causes the frontal cloud band
(with its precipitation) to appear ahead the surface front. For the kata cold front, the ascending warm
conveyor belt is overrun by the dry intrusion and tends to lead to weaker convection in general.
c. In terms of the conveyor belt model for fronts, explain the structure and development of a
warm front.
Answer: A warm front is the zone between advancing warmer, moist air and cooler dry air. The warm
conveyor belt starts behind the frontal surface in the lower levels of the troposphere, crosses the surface
front and rises to the upper levels of the troposphere. There the warm conveyor belt turns to the right
(anticyclonically) and stops rising, when the relative wind turns to a direction parallel to the front. If there
is enough humidity in the atmosphere, the result of this ascending warm conveyor belt is condensation
and more and more higher cloudiness. The cold conveyor belt in the lower layers, approaching the warm
front perpendicularly in a descending motion, turns immediately in front of the surface warm front
parallel to the surface front line. From there on the cold conveyor belt ascends parallel to the warm front
below the warm conveyor belt. Due to the evaporation of the precipitation from the warm conveyor belt
within the dry air of the cold conveyor belt, the latter quickly becomes moister and saturation may occur
with the consequence of a possible merging of the cloud systems of warm and cold conveyor belt to form
a dense nimbostratus.
4. Explain why air rotates counterclockwise around a low pressure system and clockwise around a
high pressure system in the Northern Hemisphere? Describe with the aid of a free-body diagram
how you were able to come to your conclusions.
Answer: Around a low pressure system, the pressure gradient force is directed into the low pressure
system and the Coriolis force is directed away from the low pressure system, as shown in the figure
below. Since the centripetal force is pointed inward and centripetal force is the difference between the
pressure gradient force and the Coriolis force, the pressure gradient force is larger than the Coriolis force
around a low pressure system. The intended path of air is directed in the same direction of the pressure
gradient force, which is directed towards low pressure. The Coriolis force deflects the air to the right of its
intended direction of motion in the Northern Hemisphere and thus, for low pressure, air moves
counterclockwise around a low pressure system.
The same logic can be given in reverse to describe the flow around a high pressure system in the Northern
Hemisphere. For a high pressure system, the pressure gradient force is directed outward. Since the
centripetal force is pointed inward and is the difference between the pressure gradient force and the
Coriolis force, the Coriolis force becomes larger than the pressure gradient force. The intended path of air
is directed in the same direction of the pressure gradient force, which is directed away from high pressure.
The Coriolis force deflects the air to the right of its intended direction of motion in the Northern
Hemisphere and thus air moves clockwise around a high pressure system.
5. In the Northern Hemisphere, you observe surface winds shift from N to NE to E, then to SE.
From this observation, you determine that a westerly high-pressure area (anticyclone) has passed
north of your location. Describe with the aid of a diagram how you were able to come to this
conclusion.
Answer: As winds shift from N to NE to SE, we observe that the winds are blowing clockwise. Since the
winds are blowing clockwise, then the air must be blowing parallel to isobars of a high pressure system. If
the winds are far enough above the ground such that friction is negligible, then we can assume that the
winds are in gradient balance, meaning that the wind is directed to the right of the pressure gradient force.
Therefore, a northerly wind indicates high pressure to the west. As the northerly wind shifts to an easterly
wind, the high pressure shifts from the west to the north. As the easterly wind to a southeasterly wind, the
high pressure shifts from the north to the east. Therefore, the high pressure system has passed north of
your location going towards the east.
6. Download and print the attached regional surface maps for 25 April 2008:
00Z 25 April 2008
06Z 25 April 2008
12Z 25 April 2008
For each surface map, complete the following tasks:
a. Contour surface pressure (i.e. draw isobars) at a 4-mb contour interval, using 1012 as your
reference contour. Isobars are to be drawn using a solid black line. Be sure to label your contours
and mark the locations of highs with a bold, blue H and the locations of lows with a bold, red L.
b. Contour temperature (i.e. draw isotherms) at a 5°F contour interval, using 45°F as your reference
isotherm. Isotherms are to be drawn using a dashed red line. Be sure to label your contours.
c. Shade areas of cold air advection (blue) and warm air advection (red). Explain your reasoning.
d. Identify frontal boundaries on the surface chart using all the provided information (i.e.
temperature advection, the surface temperature, pressure, wind and moisture patterns). When
drawing your frontal boundaries, be certain to use the conventional frontal symbols and colors.
Since fronts are areas of low(er) pressure, be sure to “kink” any isobars that intersect your
front(s). This should be done after you finalize the frontal positions and before you finalize your
isobars.
e. Determine whether or not the fronts have a katafront structure or an anafront structure. You may
use any other map of your choosing to supplement your answer.
After completing the contour analysis, describe the evolution of the weather.
Answer: Based on the map analysis below, we have a broad cold front moving throughout the Northern
Plains. Since the isotherms are becoming more compact over time, the cold front is also intensifying
during the 12 hour period shown. The intensification which occurs is due primarily to convergence along
the frontal boundary. As the front passes through the region, the temperatures in Colorado and Kansas
drop from the low 80s to the low 30s-40s. There is warm advection over the Midwest which is being
displaced by the movement of the cold front
7. Nielsen (1988) used data from a mesoscale observing network in New England to understand the
causes of coastal frontogenesis in New England. After reading his paper, answer the following
questions
a. What are the similarities and differences between coastal fronts and synoptic fronts?
Answer: Similar to synoptic fronts, coastal fronts are defined by their horizontal temperature gradient and
their motion is guided by the upper-level winds as well as the motion of the low-pressure system.
Contrary to synoptic fronts, coastal fronts depend on the existence of a strong land-sea temperature
contrast and significant onshore flow. This also implies the surface friction, orography, and coastal
configuration play a strong role in the development of coastal fronts.
b. Explain the three distinct types of coastal frontogenesis that are found to occur in New
England.
Answer: There are three observed classes of frontogenesis (called Types A, B, and C). Type A occurs
during the transition from offshore winds to onshore winds caused by the passage of high pressure. Type
B is triggered by nightfall as the land-sea thermal contrast establishes a coastal convergence zone. Type C
is triggered by the combination of warm air advection from a surface low and radiational cooling.
Type A coastal fronts share the following characteristics. Air temperatures before frontogenesis are colder
than SSTs. The coastal front forms quickly, within three hours, as easterly winds develop offshore. Winds
inland do not veer to easterly, but instead become parallel to the coastal front during the first few hours of
frontogenesis. The temperature difference across the front is caused by the confluence of air parcels
which have experience differential heating.
Type B coastal fronts share the following characteristics. Initially, temperatures over the coastal plain are
comparable to temperatures over water and the wind is weakly onshore. During the early evening,
radiational cooling establishes a coastal temperature gradient. As the gradient develops, winds over land
back until they are northerly. The coastal front forms in the early evening within the coastal confluence
zone. Once formed, the front resemble those resulting from Type A coastal frontogenesis.
Type C coastal fronts share the following characteristics. Air temperatures are comparable to or warmer
than SSTs. Large scale warm advection from the S/SE is occurring, often accompanied by radiational
cooling at the surface. The first manifestation of coastal frontogenesis is a sudden decrease of wind speed
or reversal of wind direction at inland stations. The temperature difference across the front is generated by
warm advection within the warmer air.
c. Would the formation mechanisms for New England coastal frontogenesis apply for
Carolina coastal frontogenesis? Defend your answer.
Answer: Type A coastal frontogenesis is unlikely to occur in the Carolinas because there usually will not
be significant cold air outbreaks near the Carolina coasts. Type B coastal frontogenesis is a likely
mechanism for Carolina coastal frontogenesis since the coastal configuration allows for the development
of a convergence zone. Type C coastal frontogenesis can also apply in the Carolinas due to the
reoccurring presence of warm advection from the South.
8. From 22-25 April 2010, a complex low pressure system moved eastward across the Great Plains,
accompanied by a strong cold front extending southwards from the system that spawned 88
tornadoes. In this problem, you are going to analysis the development of the frontal system.
According to the weather maps given to you, answer the following questions
a. From 12 Z on April 22 to 12 Z on April 23, use the frontogenesis function to analyze the
intensification of the frontal system.
Answer: The surface maps for 12 Z on April 22 is shown. Based on the frontogenesis function, we have
strong frontogenetical forcing from shear and convergence along the cold front with minimal impact from
differential diabatic heating. However, we have strong frontolysis along the warm front due to differential
diabatic heating.
The surface maps from 00 Z on April 23 is shown below. Here, we have strengthened frontogenesis from
shear and convergence, but we have frontolysis associated with differential diabatic heating along the
cold front. This overall affect will lead to a strengthening of the cold front. Due to the decrease of virtual
potential temperature in cross front direction along the warm front, shearing frontogenesis will yield
frontogenesis along the warm front with small positive contributions from convergence. This will cause
an increase in frontogenesis for both fronts.
b. From 12 Z on April 23 to 12 Z on April 24, use the frontogenesis function to analyze the
intensification of the frontal system.
Answer: On 12 Z on April 23, the strong frontogenesis by convergence is being offset by differential
diabatic heating (since surface heating will decreases the temperature gradient along the cold front).
Along the warm front, there will be frontolysis along the warm front due strong precipitation in the cold
side of the warm front and shearing. The surface maps for 00 Z on April 24 are given below
The warm front have moved northward due to increased warm air advection. The cold front will begin to
intensify as differential diabatic heating (radiative cooling will increase the temperature gradient
throughout the evening), whereas as shear frontogenesis will begin to further weaken the warm front. This
leads to strong convection along the cold front and a stationary front associated with the warm front, seen
in the figures below. There is significant frontogenesis throughout Louisiana with large gradients of
virtual potential temperature, strong convergence, and strong shearing frontogenesis.
c. From 12 Z on April 24 to 12 Z on April 25, qualitatively evaluate each term of the
frontogenesis function to analyze the weakening of the frontal system.
Answer: The strong precipitation that develops at 12 Z on April 24 within the warm sector of the storm
will eventually cause the frontal system to dissipate. The surface maps for 00 Z on April 25 are given
below. The streamlines show significant convergence along the frontal boundary, but since the advancing
air of the cold front has warmed significantly, we see a frontolytic effect reducing the horizontal
temperature gradients. Eventually, the cold front erodes the warm sector of the storm and produces a
warm occlusion. This is observed by noticing that the advancing air behind the cold front is significantly
warmer than the cold air associated with the former warm front. Therefore, the warm front undercuts the
cold front as the system occludes.
d. What type of occlusion develops as the system weakens?
Answer: Since the advancing air behind the cold front is significantly warmer than the cold air associated
with the former warm front. Therefore, the warm front undercuts the cold front and a warm occlusion
develops.
FOR PHYS 412 STUDENTS ONLY
9. A front can be thought of as a discontinuity in temperature and density. Let’s assume that the
pressure is continuous along an east-west oriented front, with cold air to the north and warm air to
the south, as shown below. Therefore, an infinitesimal change in pressure across the front can be
written mathematically as
(
)
(
)
(Note that
since this is an east-west oriented front).
a. Assuming an atmosphere that is in hydrostatic balance and assuming that the pressure is
continuous across the front, show that the slope of the front is given by
[( )
( ) ]
(
)
where the subscripts w and c denote the warm and cold sector, respectively.
Answer: Since pressure is continuous along the front, the differential vanishes at the front. Taking the
differential along the frontal boundary gives
(
)
[(
)
(
) ]
[(
)
(
Since this is an east-west oriented front, then
Substituting the hydrostatic equation for
) ]
[(
)
(
) ]
. Therefore, we have
( )
( )
( )
( )
gives
[( )
(
( ) ]
)
b. Typically, fronts slope towards the cold air such that
. Using this fact and the
expression derived in (a), explain why the pressure gradient is discontinuous across the
frontal boundary and thus, there must be a sharp point, or “kink” in the isobars along a
frontal zone.
Answer: Since fronts slope toward cold air, then
implies that
(
and by definition
)
(
. Therefore, this
)
The above equation states that the pressure gradient changes abruptly along the frontal boundary. This
implies that the pressure gradient is discontinuous along the frontal boundary and thus there must be a
kink along the frontal zone.
c. Using the expression derived in (a) and the expressions for geostrophic wind, show that
the slope of the front can be written as
[
]
(
)
Using the expression derived above, explain why there must be cyclonic shear vorticity
across a front if
. [3 points]
Answer: Substituting the geostrophic wind equations
Into the frontal slope equation gives
[
]
(
As mentioned previously,
and
)
. Therefore, this implies that
This implies that the geostrophic wind changes abruptly across the frontal boundary. This implies that
should be large across the boundary and thus, there is large shear geostrophic vorticity across the
boundary.
10. The gradient wind equation for a unit mass of air can be expressed in cylindrical coordinates as
where
is the Coriolis parameter and
is the density.
(a) In the limit of large radius of curvature (i.e. large ), show that the solution of the gradient
wind equation yields the geostrophic wind solution.
Answer: We can rewrite the gradient wind equation as
The solution of the above equation can be given by the quadratic equation
√
(
)
For large , we can apply the binomial expansion to the term in the parenthesis to obtain
(
)
(
)
The positive root gives
(
)
The negative root gives
(
)
Since the negative root is unbounded for large r, we take the positive root, which is the condition for
geostrophic balance. This makes physical sense because large r (i.e. large radius of curvature) implies that
the isobars are approximately straight.
(b) Show that the gradient wind equation produces cyclonic flow for a low pressure system
(
and
). Draw the free body diagram for this case.
Answer: The solution for gradient wind balance is
(
Since
Therefore,
and
)
, this implies that the term in the parenthesis is always greater than 1.
(
)
This implies cyclonic flow around low pressure.
(c) Show that the gradient wind equation produces anticyclonic flow for a high pressure system
(
and
) if | |
. Draw the free body diagram for this case.
Answer: The solution for gradient wind balance is
(
)
Since
, the term in the parenthesis will only produce a real value if
|
|
Substituting this restraint into the gradient wind equation gives
(
( )
)
Therefore, it follows that | |
. Therefore, gradient wind balance imposes an upper limit on the
tangential wind speed around a high-presssure system. If this constraint holds, this implies that
(
)
This implies the wind flow clockwise around a high pressure system
(d) What does the results of (b) and (c) suggests about the relative magnitude of the wind speed
around a low-pressure system compared to the relative magnitude of the wind speed around a
high-pressure system?
Answer: Since | |
for a high pressure system, gradient wind balance imposes an upper limit on
the tangential wind speed around a high-pressure system. Therefore, the relative magnitude of the wind
around a low-pressure system is higher than a high-pressure system.